János Bolyai, Non-Euclidean Geometry, and the Nature of
Space. Jeremy J. Gray. x + 185 + 54 pp. Burndy Library
Publications (distributed by the MIT Press), 2004. $20, paper.
János Bolyai, a Hungarian engineer and army officer, was one
of at least three inventors of non-Euclidean geometry. Worried that
rivals might overtake him, he hastily published his work in 1832 as
an appendix to a book by his father. It was already too late;
Nikolai Lobachevskii had published similar ideas in 1829. But Bolyai
and Lobachevskii were to share a fate even more bitter than being
scooped: They were ignored. Each of them had resolved a question
that had been nagging at mathematicians for more than 2,000 years,
yet no one paid attention.
For Bolyai, this handsome new book from the Burndy Library brings
belated vindication. It reproduces the 1832 appendix both in the
original Latin and in an English translation by George Bruce
Halsted. Perhaps more important, it includes a thoughtful
introductory essay and tutorial by Jeremy J. Gray, illustrated with
many helpful diagrams. The book, published solely in a paperback
edition, is exceptionally well designed, printed and bound.
What's non-Euclidean about Bolyai's geometry? Euclid began his work
with five postulates, or statements to be accepted without proof;
four of the postulates seemed innocuous, but many readers found the
fifth one suspect. A modern statement of this fifth postulate goes
like this: In the plane defined by a line L and a point
P not on L, exactly one line can be drawn through
P that never intersects L. Most skeptics of this
parallel postulate didn't doubt its truth but rather its necessity;
they thought it could be derived from simpler notions. Bolyai showed
something quite different: He constructed a new geometry in which
many lines through P are parallel to L.
It is the geometry of a strange place—for example, the angles
of a triangle sum to less than 180 degrees, and the Pythagorean
theorem fails—but Bolyai showed it is internally consistent
and just as logical as Euclid's formulation. (Since Einstein
introduced general relativity, the strange place described by
non-Euclidean geometry has been recognized as our own universe.)
Bolyai's story is a sad one of genius unappreciated. His appendix
apparently had only one competent reader during his lifetime, and even
that led only to more anguish. The reader was Carl Friedrich Gauss, the
most eminent mathematician of the age, who was sent a copy of the book
with the appendix. Gauss wrote back, "If I commenced by saying that
I am unable to praise this work, you would certainly be surprised for a
moment. But I cannot say otherwise. To praise it would be to praise
myself." He went on to explain that he had entertained the same
ideas for 30 or 35 years but had refrained from publishing them. Bolyai
was appalled and suspected dishonesty, but a later examination of
Gauss's notebooks confirmed that Gauss was indeed the third—or
rather the first—inventor of non-Euclidean geometry. Perhaps it's
just as well for Bolyai that he died before the notebooks were
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